Prove that if $f'(x)=f(x)$ for all $x$, then there exist a constant $c$ such that $f(x)=ce^x$ for all $x$

Question

I have the following exercise:

Prove that if $f'(x)=f(x)$ for all $x$, then there exist a constant $c$ such that $f(x)=ce^x$ for all $x$

My attempt:

Let's consider the function $g(x)=f(x)-ae^x$, $g'(x)=f'(x)-ae^x=f(x)-ae^x=0$ then $g$ is constant. Then $g(x)=k=f(x)-ae^x$, then $f(x)=ce^x$

I think it is incorrect because of how I get to the $f(x)=ce^x$ part. I would really appreciate your help with this problem, thank you.